Abstract

For any matrix automorphism group $\G$ associated with a bilinear
or sesquilinear form, Mackey, Mackey, and Tisseur have recently
shown that the matrix sign decomposition factors of $A\in\G$ also
lie in $\G$; moreover, the polar factors of $A$ lie in $\G$ if the
matrix of the underlying form is unitary. Groups satisfying the
latter condition include the complex orthogonal, real and complex
symplectic, and pseudo-orthogonal groups. This work is concerned
with exploiting the structure of $\G$ when computing the polar and
matrix sign decompositions of matrices in $\G$. We give sufficient
conditions for a matrix iteration to preserve the group structure
and show that a family of globally convergent rational
Pad\'e-based iterations of Kenney and Laub satisfy these
conditions. The well-known scaled Newton iteration for computing
the unitary polar factor does not preserve group structure, but we
show that the approach of the iterates to the group is precisely
tethered to the approach to unitarity, and that this forces a
different and exploitable structure in the iterates. A similar
relation holds for the Newton iteration for the matrix sign
function. We also prove that the number of iterations needed for
convergence of the structure-preserving methods can be precisely
predicted by running an associated scalar iteration. Numerical
experiments are given to compare the cubically and quintically
converging iterations with Newton's method and to test stopping
criteria. The overall conclusion is that the structure-preserving
iterations and the scaled Newton iteration are all of practical
interest, and which iteration is to be preferred is
problem-dependent.